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92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import OAI.Dynamics.ConditionalShuffle.InstrumentAppend
namespace OAI
noncomputable section
namespace Revealed.Instrument
open scoped Classical
open Thorp Thorp.Fourier Revealed.Disintegration
def chunks (S : Type) (t : ℕ) : (n : ℕ) → (Fin n → Fin t → S) ≃ (Fin (t*n) → S)
| 0 => by simpa only [Nat.mul_zero] using (Equiv.ofUnique (Fin 0 → Fin t → S) (Fin 0 → S))
| n+1 => ((Fin.snocEquiv (fun _ : Fin (n+1) => Fin t → S)).symm.trans
(Equiv.prodComm _ _)).trans ((Equiv.prodCongr (chunks S t n) (Equiv.refl _)).trans
(Fin.appendEquiv (t*n) t))
lemma chunks_snoc {S : Type} (t n : ℕ) (p : Fin n → Fin t → S) (q : Fin t → S) :
chunks S t (n+1) (Fin.snoc p q) = Fin.append (chunks S t n p) q := by
simp [chunks, Fin.appendEquiv]
variable {E S G Ω : Type} [Fintype S] [Fintype G] [Group G] [Fintype Ω] [Nonempty Ω]
variable (I : Data E S G Ω)
lemma block_base (e : E) (t n : ℕ) (p : Fin n → Fin t → S) :
base (block I t) e n p = base I e (t*n) (chunks S t n p) := by
induction n with
| zero => rfl
| succ n ih =>
obtain ⟨⟨q,p⟩,rfl⟩ := (Fin.snocEquiv (fun _ : Fin (n+1) => Fin t → S)).surjective p
erw [base_snoc, chunks_snoc]
change base I (base (block I t) e n p) t q = base I e (t*n+t) (Fin.append (chunks S t n p) q)
rw [base_append, ih]
lemma block_weights (e : E) (t n : ℕ) (p : Fin n → Fin t → S) :
weights (block I t) e n p = weights I e (t*n) (chunks S t n p) := by
induction n with
| zero => rfl
| succ n ih =>
obtain ⟨⟨q,p⟩,rfl⟩ := (Fin.snocEquiv (fun _ : Fin (n+1) => Fin t → S)).surjective p
erw [weights_snoc, chunks_snoc, block_probability, block_base, ih]
exact (weights_append I e (t*n) t (chunks S t n p) q).symm
lemma block_product (e : E) (t n : ℕ) (p : Fin n → Fin t → S)
(hp : weights (block I t) e n p ≠ 0) :
realProduct n (kernels (block I t) e n p) =
realProduct (t*n) (kernels I e (t*n) (chunks S t n p)) := by
induction n with
| zero => rfl
| succ n ih =>
obtain ⟨⟨q,p⟩,rfl⟩ := (Fin.snocEquiv (fun _ : Fin (n+1) => Fin t → S)).surjective p
erw [weights_snoc] at hp
have hp0 := (mul_ne_zero_iff.mp hp).1
have hq := (mul_ne_zero_iff.mp hp).2
rw [block_probability] at hq
erw [kernels_snoc, realProduct_snoc, ih p hp0, chunks_snoc]
change realConv (conditional (pathJoint I (base (block I t) e n p) t) q)
(realProduct (t*n) (kernels I e (t*n) (chunks S t n p))) =
realProduct (t*n+t) (kernels I e (t*n+t) (Fin.append (chunks S t n p) q))
rw [conditional_pathJoint I _ t q hq, block_base, kernels_append, realProduct_append]
theorem distance_block (e : E) (t n : ℕ) : distance (block I t) e n = distance I e (t*n) := by
unfold distance
rw [← Equiv.sum_comp (chunks S t n)]
apply Finset.sum_congr rfl
intro p _
by_cases hp : weights (block I t) e n p = 0
· rw [← block_weights, hp, zero_mul, zero_mul]
· rw [block_product I e t n p hp, block_weights]
end Revealed.Instrument
end
end OAI
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